Lemma 25.9.1. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $L$ be a simplicial object of $\text{SR}(\mathcal{C}, X)$. Let $n \geq 0$. Consider the commutative diagram

25.9.1.1
$$\label{hypercovering-equation-diagram} \xymatrix{ \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L)_{n + 1} \ar[r] \ar[d] & (\text{cosk}_ n \text{sk}_ n \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L))_{n + 1} \ar[d] \\ (L \times L)_{n + 1} \ar[r] & (\text{cosk}_ n \text{sk}_ n (L \times L))_{n + 1} }$$

coming from the morphism defined above. We can identify the terms in this diagram as follows, where $\partial \Delta [n + 1] = i_{n!}\text{sk}_ n \Delta [n + 1]$ is the $n$-skeleton of the $(n + 1)$-simplex:

\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L)_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits (\Delta [1] \times \Delta [n + 1], L)_0 \\ (\text{cosk}_ n \text{sk}_ n \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L))_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits (\Delta [1] \times \partial \Delta [n + 1], L)_0 \\ (L \times L)_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits ( (\Delta [n + 1] \amalg \Delta [n + 1], L)_0 \\ (\text{cosk}_ n \text{sk}_ n (L \times L))_{n + 1} & = & \mathop{\mathrm{Hom}}\nolimits ( \partial \Delta [n + 1] \amalg \partial \Delta [n + 1], L)_0 \end{eqnarray*}

and the morphism between these objects of $\text{SR}(\mathcal{C}, X)$ come from the commutative diagram of simplicial sets

25.9.1.2
$$\label{hypercovering-equation-dual-diagram} \xymatrix{ \Delta [1] \times \Delta [n + 1] & \Delta [1] \times \partial \Delta [n + 1] \ar[l] \\ \Delta [n + 1] \amalg \Delta [n + 1] \ar[u] & \partial \Delta [n + 1] \amalg \partial \Delta [n + 1] \ar[l] \ar[u] }$$

Moreover the fibre product of the bottom arrow and the right arrow in (25.9.1.1) is equal to

$\mathop{\mathrm{Hom}}\nolimits (U, L)_0$

where $U \subset \Delta [1] \times \Delta [n + 1]$ is the smallest simplicial subset such that both $\Delta [n + 1] \amalg \Delta [n + 1]$ and $\Delta [1] \times \partial \Delta [n + 1]$ map into it.

Proof. The first and third equalities are Simplicial, Lemma 14.17.4. The second and fourth follow from the cited lemma combined with Simplicial, Lemma 14.21.11. The last assertion follows from the fact that $U$ is the push-out of the bottom and right arrow of the diagram (25.9.1.2), via Simplicial, Lemma 14.17.5. To see that $U$ is equal to this push-out it suffices to see that the intersection of $\Delta [n + 1] \amalg \Delta [n + 1]$ and $\Delta [1] \times \partial \Delta [n + 1]$ in $\Delta [1] \times \Delta [n + 1]$ is equal to $\partial \Delta [n + 1] \amalg \partial \Delta [n + 1]$. This we leave to the reader. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).